Functional Analysis Lecture Notes: Quotient Spaces
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چکیده
Example 1.1 (Cosets in R). Consider the vector space X = R. Let M be any onedimensional subspace of R, i.e., M is a line in R through the origin. A coset of M is a rigid translate of M by a vector in R. For concreteness, let us consider the case where M is the x1-axis in R , i.e., M = {(x1, 0) : x1 ∈ R}. Then given a vector y = (y1, y2) ∈ R , the coset y +M is the set y +M = {y +m : m ∈ M} = {(y1 + x1, y2 + 0) : x1 ∈ R} = {(x1, y2) : x1 ∈ R}, which is the horizontal line at height y2. This is not a subspace of R , but it is a rigid translate of the x1-axis. Note that there are infinitely many different choices of y that give the same coset. Furthermore, we have the following facts for this particular setting.
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